Gorenstein Symmetry Conjecture: does one-sided finiteness of the self-injective dimension force the other side for Artin algebras and two-sided noetherian rings?

The Gorenstein Symmetry Conjecture (GSC) of Auslander–Reiten predicts that for an Artin algebra, finiteness of the injective dimension of the regular module on one side implies finiteness on the other side, in which case the two self-injective dimensions coincide. A preprint of 30 July 2026 (arXiv:2607.28011) claiming a complete homological proof was withdrawn by its own author one day later with an acknowledged error, so the conjecture stands open. The same symmetry question is equally open, with no known counterexamples, for arbitrary left-and-right noetherian rings; every known positive result imposes additional hypotheses — the Auslander condition, existence of a dualizing complex, or membership in special classes such as weakly Gorenstein algebras. As of August 2026, even the Auslander-condition version (Huang's question for general two-sided noetherian rings) is resolved only for module-finite algebras over commutative noetherian rings. The open cases are the Artin-algebra setting and the unconditional two-sided noetherian setting.

Background

An Artin algebra is a ring $\Lambda$ that is finitely generated as a module over a commutative artinian center; finite-dimensional algebras over a field are the basic examples, and every Artin algebra is two-sided artinian, hence two-sided noetherian. For a left module $M$ over a ring $R$, the injective dimension $\mathrm{idim}_{R}M$ is the minimal length of an injective resolution of $M$ (infinite if no finite one exists); the injective dimensions of $\Lambda$ as a left and as a right module over itself are called its left and right self-injective dimensions, and nothing in the elementary theory forces them to behave symmetrically. An Artin algebra is called Gorenstein (in the sense of Happel and Auslander–Reiten) if its regular module has finite injective dimension on both sides; Happel's Gorenstein homological algebra — Gorenstein projective modules, singularity categories, (virtually) Gorenstein algebras — is built on this two-sided finiteness. A classical lemma of Zaks for semiprimary rings and a lemma of Auslander–Reiten for Artin algebras show that whenever both self-injective dimensions are finite they are equal, so the only genuine question is whether one-sided finiteness implies the other side. The Gorenstein Symmetry Conjecture (GSC), formulated by Auslander and Reiten in the early 1990s, asserts exactly this for all Artin algebras; it is one of the classical homological conjectures of the representation theory of algebras, and no Artin algebra with unequal (finite-versus-infinite) self-injective dimensions has ever been exhibited. The question extends verbatim to a left and right noetherian ring $A$: if $\mathrm{idim}_{A}A<\infty$, must $\mathrm{idim}_{A^{\mathrm{op}}}A<\infty$ (equality then holding as before)? Shaul records that this noetherian version is open and that no counterexample is known. Two families of partial results delimit current knowledge. First, under the Auslander condition — defined for a two-sided noetherian ring by requiring that in a minimal injective coresolution of the regular module, the $i$-th term has flat dimension at most $i$ for every $i$ — Auslander and Reiten proved the symmetry implication for Artin algebras, Huang extended it to two-sided artinian rings and posed the same question for general two-sided noetherian rings, and Shen (August 2026) settled it for module-finite algebras over commutative noetherian rings, while explicitly noting that his counting step has no evident analogue for an arbitrary two-sided noetherian ring. Second, under dualizing-complex hypotheses — a dualizing complex is a bounded complex of bimodules with finite injective dimension on both sides whose derived Hom into it duality fixes the derived category — Shaul gave several necessary and sufficient criteria for symmetry to hold, but the existence of a dualizing complex is itself an unverified hypothesis, so this is a characterization rather than a resolution. For Artin algebras unconditionally, the conjecture is verified only inside special classes: algebras realizable as endomorphism algebras over representation-finite algebras — algebras with only finitely many isomorphism classes of indecomposable finitely generated modules — (Wei), left weakly Gorenstein algebras, i.e. algebras for which every finitely generated module of finite injective dimension has finite projective dimension (Ringel–Zhang and Marczinzik), and it can be reduced by recollements of derived categories — a recollement is a glueing of one triangulated category between two others, here relating the derived categories of three algebras — (Han–Qin), by excellent extensions (Y. Zhang), and by eventually homological isomorphisms (Qin); Becerril (2026) confirms it when every left injective has finite flat dimension or when the projectives have finite strongly FP-injective dimension. A preprint of 30 July 2026 by Weiqing Li claiming a complete homological proof was withdrawn by the author on 31 July 2026 with the comment that there is an error in the proof, which restores the conjecture's open status, retained in the recent literature from Chen's 2017 survey to Becerril's February 2026 paper.

Problem Statement

Let $A$ be a left and right noetherian ring (in particular, let $\Lambda$ be an Artin algebra, the classical setting of the Gorenstein Symmetry Conjecture of Auslander–Reiten). Suppose that the injective dimension of the regular module is finite on one side, say $\mathrm{idim}_{A}A<\infty.$ Does it follow that the injective dimension on the other side is finite, $\mathrm{idim}_{A^{\mathrm{op}}}A<\infty\ ?$ (When both sides are finite the two dimensions are equal — Zaks 1969 for semiprimary rings, Auslander–Reiten 1991 for Artin algebras — so an affirmative answer makes the self-injective dimension an unambiguous two-sided invariant.) The Artin-algebra case and the unconditional two-sided noetherian case are both open, and no ring with one side finite and the other infinite is known. The closely related sub-question of Huang, also unresolved, asks whether the implication holds for an arbitrary two-sided noetherian ring satisfying the Auslander condition; it is currently known for Artin algebras (Auslander–Reiten 1994), two-sided artinian rings (Huang 2007), and module-finite algebras over commutative noetherian rings (Shen 2026), but open in the stated generality, and any answer addressing only that restricted sub-question resolves a strictly weaker statement than the conjecture itself.

References

1. Maurice Auslander, Idun Reiten, 'Applications of contravariantly finite subcategories', Advances in Mathematics 86 (1991), 111–152. DOI: 10.1016/0001-8708(91)90037-8, https://doi.org/10.1016/0001-8708(91)90037-8
2. Maurice Auslander, Idun Reiten, 'k-Gorenstein algebras and syzygy modules', Journal of Pure and Applied Algebra 92 (1994), 1–27. DOI: 10.1016/0022-4049(94)90044-2, https://doi.org/10.1016/0022-4049(94)90044-2
3. Abraham Zaks, 'Injective dimension of semi-primary rings', Journal of Algebra 13 (1969), 73–86. DOI: 10.1016/0021-8693(69)90007-6, https://doi.org/10.1016/0021-8693(69)90007-6
4. Jiaqun Wei, 'Finitistic and Representation Dimensions', arXiv:0803.3364 (2008), https://arxiv.org/abs/0803.3364
5. Yang Han, Yongyun Qin, 'Reducing homological conjectures by n-recollements', arXiv:1410.3223 (2014), https://arxiv.org/abs/1410.3223
6. Zhaoyong Huang, 'Generalized tilting modules with finite injective dimension', Journal of Algebra 311 (2007), 619–634. DOI: 10.1016/j.jalgebra.2006.11.025, https://doi.org/10.1016/j.jalgebra.2006.11.025
7. Xiao-Wu Chen, 'Gorenstein Homological Algebra of Artin Algebras', arXiv:1712.04587 (2017), https://arxiv.org/abs/1712.04587
8. Yingying Zhang, 'Excellent extensions and homological conjectures', arXiv:1702.05902 (2017), https://arxiv.org/abs/1702.05902
9. Claus Michael Ringel, Pu Zhang, 'Gorenstein-projective and semi-Gorenstein-projective modules', Algebra & Number Theory 14 (2020), no. 1, 1–36. DOI: 10.2140/ant.2020.14.1, https://arxiv.org/abs/1808.01809
10. Yongyun Qin, 'Eventually homological isomorphisms and Gorenstein projective modules', arXiv:2203.09211 (2022), https://arxiv.org/abs/2203.09211
11. Liran Shaul, 'Acyclic complexes of injectives and finitistic dimensions', arXiv:2303.08756 (2023), https://arxiv.org/abs/2303.08756
12. Zhaoyong Huang, 'On Auslander-Type Conditions of Modules', Publications of the Research Institute for Mathematical Sciences 59 (2023), no. 1, 57–88. DOI: 10.4171/prims/59-1-2, https://doi.org/10.4171/prims/59-1-2
13. Víctor Becerril, 'Strongly $FP$-injective dimensions and Gorenstein projective precovers', arXiv:2602.00932 (2026), https://arxiv.org/abs/2602.00932
14. Weiqing Li, 'On a proof of the Gorenstein Symmetry Conjecture', arXiv:2607.28011 (2026; v2 of 31 July 2026 withdrawn by the author with the comment 'Since there is an error in the proof, I hereby withdraw the article'), https://arxiv.org/abs/2607.28011
15. Dawei Shen, 'Symmetry of self-injective dimensions under the Auslander condition', arXiv:2608.15974 (2026), https://arxiv.org/abs/2608.15974